Along similar lines, recall there are self-diffeomorphisms of the disc \(\mathbb {D}^n\)  that are not isotopic to the identity, as long as \(n\ge 5\)  [1]}. Using this, one may find a self-diffeomorphism of \(\mathcal {C}_{d,r}(O,Y)\) , supported on a small disc, that is not properly isotopic to the identity [2]}; however, this diffeomorphism will be isotopic to the identity by “pushing its support to infinity” in \(\mathcal {C}_{d,r}(O,Y)\) . We are grateful to Oscar Randal-Williams for explaining this to us.

